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Title

ON LOCAL ANTIMAGIC CHROMATIC NUMBER OF GRAPHS

Writers

Shaebani Saeed

Pages

 Start Page 245 | End Page 256

Abstract

 A {\it local Antimagic labeling} of a connected graph GG with at least three vertices, is a bijection f: E(G)→ {1, 2, … , |E(G)|}f: E(G)→ {1, 2, … , |E(G)|} such that for any two adjacent vertices uu and vv of GG, the condition ω f(u)≠ ω f(v)ω f(u)≠ ω f(v) holds; where ω f(u)=∑ x∈ N(u)f(xu)ω f(u)=∑ x∈ N(u)f(xu). Assigning ω f(u)ω f(u) to uu for each vertex uu in V(G)V(G), induces naturally a proper vertex coloring of GG; and |f||f| denotes the number of colors appearing in this proper vertex coloring. The {\it Local antimagic chromatic number} of GG, denoted by χ la(G)χ la(G), is defined as the minimum of |f||f|, where ff ranges over all local Antimagic labelings of GG. In this paper, we explicitly construct an infinite class of connected graphs GG such that χ la(G)χ la(G) can be arbitrarily large while χ la(G∨ K2¯ )=3χ la(G∨ K2¯ )=3, where G∨ K2¯ G∨ K2¯ is the join graph of GG and the complement graph of K2K2. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex coloring of a graph, Graphs and Combinatorics 33} (2017), 275-285].

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